On the 737NG you can fool the aircraft into thinking that engine #1 is shut down to operate the LGTU. We do this on airtests, at a safe altitude, to test the LGTU by pulling the ENG 1 RUN/PWR c/b (P18-2 B3 or thereabouts) and then retract the gear with the hyd A system off. There is no equivalent c/b on the classics and I would not recommend trying to find one, god knows what else you would inadvertently take out in the process.
Re the performance issue of one engine and the gear down; I can tell you from experience that even in a 20K 737-300 at about 44,000kg in the above config and flap 15, can just maintain level flight at 15,000ft – but it wont climb!
I have been given this no hyd A, gear down scenario in the sim myself and knew that by shutting down eng #1 (or pulling the c/b) I could get the gear up, but I have declined to do it due to the increased risk to the pax. Now that you have the knowledge it is between you, your training dept & your conscience to decide what to do given your particular circumstances.
S&L